Match the following elements of with their cofactors and choose the correct answer ElementCofactor(I)– 1(a)– 2(II)1(b)32(III)3(c)4(IV)6(d)6(e)

Engineering

Mathematics

Introduction to Determinants

Properties of Determinant

Question

Match the following elements of $∣ \begin{array}{ccc} 1 & - 1 & 0 \\ 0 & 4 & 2 \\ 3 & - 4 & 6 \end{array} ∣$ with their cofactors and choose the correct answer

	Element		Cofactor
(I)	– 1	(a)	– 2
(II)	1	(b)	32
(III)	3	(c)	4
(IV)	6	(d)	6
		(e)	– 6

I - b, II - d, III - c, IV - a

I - d, II - b, III - a, IV - c

I - b, II - d, III - a, IV - c

I - d, II - a, III - b, IV - c

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$∣ \begin{array}{ccc} 1 & - 1 & 0 \\ 0 & 4 & 2 \\ 3 & - 4 & 6 \end{array} ∣$

(A) Cofactor of

- 1 = c_{12} = - ∣ \begin{array}{cc} 0 & 2 \\ 3 & 6 \end{array} ∣ = - (0 \times 6 - 2 \times 3) = - (0 - 6) = 6

(B) Cofactor of

1 = c_{11} = ∣ \begin{array}{cc} 4 & 2 \\ - 4 & 6 \end{array} ∣ = (4 \times 6 - (- 4) \times 2) = 24 + 8 = 32

3 = c_{31} = ∣ \begin{array}{cc} - 1 & 0 \\ 4 & 2 \end{array} ∣ = (- 1 \times 2 + 4 \times 0) = - 2 + 0 = - 2

(D) Cofactor of

6 = c_{33} = ∣ \begin{array}{cc} 1 & - 1 \\ 0 & 4 \end{array} ∣ = (1 \times 4 - 0 (- 1)) = 4 - 0 = 4 .